Intro to Set Theory The Math of Santa’s Bag

But first...

Now back to our regularly scheduled programming...

What is a set? (from Wikipedia) Definition: A set is a well-defined collection of distinct objects. The objects that make up a set (also known as the elements or members of a set) can be anything: numbers, people, letter of the alphabet, other sets, and so on. One of the founders of set theory Georg Cantor, gave the following definition: “A set is a gathering into a whole of definite, distinct objects of our perception or of our thought-which are called elements of the set.”

But really… what is a set? A set is a magic bag that can contain anything or nothing...

Familiar examples of sets Natural numbers (counting numbers) Integers (counting numbers and negatives) Rationals (fractions) Real numbers (fractions plus weird ones like pi etc.)

Write these to mean it’s Writing a set: First, we can list the elements: Write these to mean it’s a set

Writing a set: (no repetition) We don’t stutter when we write a set!

Writing a set: (ORDER IS NOT IMPORTANT)

Cardinality: We denote the number of elements of a set as

Example 3

Another Example: Special set: The Empty Set Look its empty!

Notation for sets: We will use capital letters to indicate sets Like: We also have a symbol for membership that is when x is an element of a set A we write:

Fun example: Pizza is in this set so we write: and say “pizza is in A”

In our Familiar Examples:

Notation for “not-in” We also have notation for not being a member of the set:

Fun example: Bob is not in this set so we write: and say “Bob is not in A”

Logic is back in the game! Membership of a set is a “statement” that is it has a truth value: This statement is true/false!

Subsets Definition: A subset is made of only the elements of another set. If A is a subset of B we write:

Subsets (logic definition) is true whenever the following statement is true for every x in A:

Weird conclusions... The empty set is a subset of any set:

There is nothing in the empty set!! WTF? Well is this statement true? This statement is FALSE!! There is nothing in the empty set!!

Universe (of discourse) We at times discuss the subsets of a given “larger set” this set is referred as The Universe of discourse, If this set doesn’t already have a name we usually denote it as:

Writing a set: “set builder notation” We can write conditions... “The set” “The statement p(x) is true” “Containing x’s” “Such that”

Example We can write the set of all even numbers:

Example We can write the set of real numbers whose square is less than 10:

Group Work: The set of real numbers bigger than 10 and less than 42 Use set builder notation to write the following sets: The set of real numbers bigger than 10 and less than 42 The set of prime natural numbers The set of of all real numbers whose cube is bigger than or equal to 27 The set of rational numbers whose denominator is 3

The Union: Definition: The Union of two sets A, B is Recall this means “OR” So it’s all the stuff in A or B…. so if it’s in A it’s in the union, also if it’s in B it must be in the union..

Let’s write the sets? First, Andy’s set:

Let’s write the sets? Now, Bob’s set:

Let’s see it:

The intersection: Definition: The intersection of two sets, is the set of elements that they share...

Let’s see it:

Group Work: (HP) (GR) (RC) (SL)

Complements (everything not in!) Definition: The complement of a set is the set of everything that’s in the universe but not the original set

Example

Group Work: 7 Given the universe above, find the complement of the following sets: 7

Venn Diagrams

Venn Diagrams (Bob and Andy)

Venn Diagrams (Union)

Venn Diagrams (Intersection)

Venn Diagrams (Complement)

Venn Diagrams (All Together)

Group Work: Shade in the indicated sets in a Venn Diagram for the following: